Article
Ufa Mathematical Journal
Volume 10, Number 4, pp. 40-49
On uniqueness of weak solution to mixed problem for integro-differential aggregation equation
Vil'danova V.F.
DOI:10.13108/2018-10-4-40
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In a well-known paper by A.Bertozzi, D.Slepcev (2010), there was established
the existence and uniqueness of solution to a mixed problem for
the aggregation equation
$$
u_t - \Delta A(x, u) + {\rm div}\, (u\nabla K \ast u)=0
$$
describing the evolution of a colony of bacteria in a bounded convex domain $\Omega$. In this paper we prove the existence and uniqueness of
the solution to a mixed problem for a more general equation
$$
\beta(x,u)_t={\rm
div}\,(\nabla A(x,u)-\beta(x,u)G(u))+f(x,u).
$$
The term $f(x,u)$ in
the equation models the processes of ``birth-destruction'' of
bacteria. The class of integral operators $G(v)$ is wide enough
and contains, in particular, the convolution operators $\nabla K \ast u$. The vector kernel $g (x,y)$ of the operator $G(u)$ can have singularities.
Proof of the uniqueness of the solution in the work by A.Bertozzi, D.Slepcev was based on the conservation of the mass $\int_\Omega u(x,t)dx=const$ of bacteria and employed the convexity of $\Omega$ and the properties of the convolution operator. The presence
of the ``inhomogeneity'' $f(x,u)$ violates the mass conservation. The proof of
uniqueness proposed in the paper is suitable for a nonuniform
equation and does not use the convexity of $\Omega$.